Math topic中文

Graphical methods

Solving inequalities by graphical methods.

Practice questions

1A curve $C_1$ has the equation $p^2x^2 - y^2 = p^2$ where $p > 1$ . Sketch $C_1$ , stating the coordinates of any points o…
2Sketch, on the same diagram, the graphs of $y = |2x + 3|$ and $y = -2x^2 - 8x - 3$ . (You are not required to label any a…
3On a single diagram, sketch the graphs of $y = |2x - p|$ and $y = qx$ where the following conditions are satisfied, indi…
4Sketch, on the same axes, the graphs of $y=5-\dfrac{1}{x-1}$ and $y=\dfrac{1}{(x-5)^2}$ . Solve the inequality $5-\dfrac{1}{x-1}<\dfrac{1}{(x-5)^2}$ …
5The curve $C$ has equation $y = \frac{x^2 + kx + 4}{x + 1}$ , where $k$ is a constant. Find the range of values of $k$ if…
6Find the exact roots of the equation $\left|\dfrac{4x-5}{x-2}\right| = 3x. On the same axes, sketch the curves with equations$ …
7(1)二进制数学 $1101101_{(2)}$ 转化为十进制数为___. (2)若实数 $x$ ， $y$ 满足约束条件 $\left\{\begin{array}{l}x\geqslant 0\\ y\leqslant x\\ 2x+y-9\leqslant 0\end{array}\right.$ …
8A curve $C$ has equation $y = \frac{x^2 - 4x + 4}{x - 1}$ . Sketch $C$ , stating clearly the equations of any asymptotes,…
9Sketch the graphs of $y = 3e^x$ and $y = x + 3$ on the same diagram. Indicate clearly the coordinates of the points of i…
10The curves $C_1$ and $C_2$ have equations $y = \frac{2x^2 + 9}{x^2 - 4}$ and $\frac{y^2}{25} - \frac{x^2}{9} = 1$ respec…
11In this question, $a$ , $b$ , $c$ and $d$ are non-zero constants. The curve $C$ has equation $y = \frac{ax^2 + bx - 7}{x - c}$ …
12The complex number $z$ satisfies $\arg(z-1-2i)=\theta$ , where $\theta$ is a fixed angle in the interval $-\pi < \theta \le \pi$ …
13It is given that $\mathrm{f}(x) = x + \frac{m^2}{x-2}$ , where $0 < m < 1$ . Sketch the graph of $y = \mathrm{f}(x)$ , show…
14It is given that $f(x) = \begin{cases} \tfrac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \tfrac{1}{4}(-3x+14), & \text{for } 2 \le x < 4, \end{cases}$ …
15Complete the table of values for $y = x^2 + 2x - 1$ On the grid opposite, draw the graph of $y = x^2 + 2x - 1$ for $-4 \le x \le 2$ …
16A curve $C$ has equation $y = 2ax + \frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x \ne 0$ . Using diff…
17Find, in terms of $a$ , the roots of the equation $\dfrac{1}{(x-a)^2}=|x-a|$ . On the same axes, sketch the curves with eq…
18When Mr. Lee sends a text message to any of his students over the weekend, he gets a reply, on average, about 6 out of 1…
19Sketch the curve with equation $y = \ln(x+1), x > -1$ , stating the equation of asymptote and intercepts with the axes. O…
20The curve C has equation $y=\dfrac{2x^{2}-a}{x+k}$ where $a,\; k>0$ . Given that $2k^{2}\ne a$ , find the range of values…
21The curves $C_{1}$ and $C_{2}$ have equations $\dfrac{x^{2}}{9} - \dfrac{y^{2}}{4} = 1$ and $y^{2} + x^{2} = k^{2}$ resp…
22(Q2) [Cambridge/9231/2007] The curve C has equation $y = \frac{ax^2 + bx + c}{x + 4},$ where $a$ , $b$ and $c$ are cons…
23On the same axes, sketch the graphs of $y = \frac{1}{x - 2}$ and $y = 4x - 2$ , indicating clearly the equations of the a…
24The curve $C$ has equation $y = \frac{x^2 + 4x + \lambda - 5}{x + r},$ where $\lambda$ is a non-zero constant, and a v…
25Sketch, on the same diagram, the curves with equations $y = \frac{|x + 1|}{|x - 2|}$ and $y = \ln\left(1 - \frac{x}{2}\right) + \tfrac{1}{2}$ …
26Sketch the curve $C$ with equation $y = \frac{x-2}{x-1}$ , stating the exact coordinates of any points of intersection wi…
27The diagram shows the graphs of $y = |a - bx|$ and $y = |mx - 6|$ where $a$ , $b$ and $m$ are positive constants. The gra…
28Sketch the graph of $(y+2)^2 = 4(x-1)^2 - 9$ , showing its main relevant features. Hence find the range of values of $k$ …
29Sketch the graph of $(y+2)^2 = 4(x-1)^2 - 9$ , showing its main relevant features. Hence find the range of values of $k$ …
30Given that both $a$ and $b$ are positive constants, on the same diagram, sketch the graphs of $y = \frac{2}{|x - a|} \quad \text{and} \quad y = b|x - a|.$ …
31On the same axes, sketch the graphs of $y = \sin 2x$ and $y = \cos x$ for $0 \le x \le \frac{\pi}{2}$ , stating the exact…
32The function f is defined by $f:x\mapsto\dfrac{1}{2-x^2},\quad x\in\mathbb{R},\;x\le 0,\;x\ne -\sqrt{2}\,.$ Define the…
33The complex number $z$ satisfies the relations $|z|\le 2$ and $\dfrac{\pi}{6}\le \arg z \le \dfrac{\pi}{2}$ . Illustrate…
34The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$ , where $a$ is a constant, $a \ne 6$ . Find the set of values…
35The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \cos\left(\frac{\pi x}{2}\right)$ , $x \in \mathbb{R}$ , $-2 < x \le 0$ …
36The hyperbola $C_1$ has equation $\frac{(x-2)^2}{3^2} - \frac{y^2}{2^2} = 1$ . Sketch $C_1$ , labelling the equations of t…
37The curves $C_1$ and $C_2$ have equations $y=\dfrac{x-b}{x-a}$ and $y=\dfrac{x-b}{b}$ respectively, where $a$ and $b$ ar…
38Sketch the graphs of $y=3e^{x}$ and $y=x+3$ on the same diagram. Indicate clearly the coordinates of the points of inter…
39Sketch, on the same diagram, the curves with equations $y = \ln(x-2)$ and $y = \frac{1}{\text{e}^x} + 3$ , stating the eq…
40Sketch the graph of $(x-2)^2 + 4(y+2)^2 = 16$ , stating the coordinates of all the vertices. On the same diagram, sketch…
41Sketch the graphs of $y = 1 + \frac{a-2}{x-a}$ and $y = -\frac{1}{a}x + \frac{2}{a}$ on a single diagram, where $a$ is a…
42The curve $C$ has equation $y=\frac{ax^2-2ax+a-2}{x-2},\quad a\in\mathbb{R},\ a\ne 0.$ Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ …
43Without using a calculator, solve the inequality $\frac{2x^2 + 3x}{2x^2 + x - 1} \le \frac{1}{2x + 2}$ . Using your answe…
44The diagram shows the graphs of $y = |a - b x|$ and $y = |m x - 6|$ where $a, b$ and $m$ are positive constants. The gra…
45The diagram shows the graphs of $y = |a - bx|$ and $y = |mx - 6|$ where $a$ , $b$ and $m$ are positive constants. The gra…
46The graphs of $y = \mathrm{f}'(x)$ and $y = |\mathrm{f}(x)|$ are shown below. State the nature of all turning point(s) o…
47The parametric equations of a curve $C_1$ are $x=3t^3+2t^2$ and $y=-t^3+2t^2+1$ for $-1 \le t \le 1$ . Another curve $C_2$ …
48Let $y = \tan\left(2\tan^{-1}x + \frac{\pi}{4}\right)$ . Show that $(1+x^2)\frac{\mathrm{d}y}{\mathrm{d}x} = 2(1+y^2)$ . H…
49[It is given that the area of a sector of a circle with radius $r$ and angle $\theta$ is given by $\frac{1}{2}r^2\theta$ …
50Sketch the graphs of $y = 1 + \frac{a - 2}{x - a}$ , and $y = -\frac{1}{a}x + \frac{2}{a}$ on a single diagram, where $a$ …
51The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below. State the nature of all turning point(s) of the graph of $y=f(x)$ …
52On the same axes, sketch the curves with equations $y = \left| \frac{ax + 1}{1 - ax} \right|$ and $y = \frac{1}{a}x + 1$ …
53Sketch, on the same diagram, the graphs of $y = \ln(2x+9)$ and $y = \sqrt{10-x^2}$ , including the coordinates of the poi…
54The function f is defined by $f(x)=ax^3+bx^2+cx+d$ , where $a,b,c$ and $d$ are real numbers. Given that $1-2i$ and $-1$ a…
55The curve $C$ has equation $y = \frac{x^2 + ax + b}{x + 1}$ , where $a$ and $b$ are constants. It is given that $C$ has a…
56The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f} : x \mapsto x^2 + 2x + 4, \quad x \le -1 \quad \text{and} \quad \mathrm{g} : x \mapsto \ln(2x - 1), \quad x > \frac{1}{2}.$ …
57Without using a calculator, solve the inequality $\dfrac{2x^2 - x}{x^2 + 3x - 4} > 1$ . On the same axes, sketch the grap…
58The complex number $z$ satisfies the relations $|z - 2\mathrm{i}| < 4$ and $|z + 4 - 6\mathrm{i}| \le |z - 2\mathrm{i}|$ …
59The complex number $z$ satisfies the following inequalities: $|z|^2 \le 4 \quad \text{and} \quad -\frac{\pi}{6} \le \arg(z + \sqrt{3} - \mathrm{i}) \le 0.$ …
60The curve C is defined by the parametric equations $x = \cos t + \dfrac{1}{2} \cos 5t, \quad y = \sin t + \dfrac{1}{2} \sin 5t \quad$ …